Zero in the Spectrum

Abstract

In the previous chapter we noted that in our study of semilinear elliptic partial differential equations of the form

$$\displaystyle \mathcal A u = f(x,u), u \in D $$

in unbounded domains, we required that the resolvent set of $\mathcal A$ not be empty. For convenience, we assumed that it contain 0. This allowed us to choose an interval $(a,b) \subset \rho (\mathcal A),$ where a < 0 < b. We let $D = D(|\mathcal A|{ }^{(1/2)}).$ With the scalar product $(u,v)_D= (|\mathcal A|{ }^{(1/2)}u,|\mathcal A|{ }^{(1/2)}v),$ it became a Hilbert space. We let

$$\displaystyle N= E(-\infty , a], \quad M = E[b, \infty ) $$

be the negative and positive invariant subspaces of $\mathcal A.$ Then

$$\displaystyle (\mathcal A v,v) \le a \|v\|{ }^2, \quad v \in D \cap N, $$

and

$$\displaystyle (\mathcal A w,w) \ge b\|w\|{ }^2, \quad w \in D \cap M. $$

The hypotheses of our theorems depended on a and b using the fact that 0 was embedded in $\rho (\mathcal A).$ The purpose of the present chapter is to study the situation when 0 is a boundary point of $\rho (\mathcal A)$ and the arguments do not work.